Why is it easier for you to make sharp turns more quickly when your center of gravity is over the handle bar?

The force that causes you to turn is friction between the front wheel and the ground. When you turn left, friction pushes the front wheel left and you turn left. By putting all of your weight over the front wheel, you accomplish two things. First, you increase the maximum static frictional force between the ground and the front wheel. You push them together harder so that they are less likely to slide (skid). Second, you make it easier for that sideways friction force to accelerate you; the force acts closer to your body and more directly on you. There are fewer torques on the bicycle that might cause it to skid about on either the front or rear wheel.